A.Z. Broder, A.M. Frieze, E. Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas, Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, Austin, TX 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that a sequence of events En occurs almost certainly if Pr[E n ] tends to 1 as n 1. For k = 2, a sharp threshold was proved in [9, 16] a random instance of 2 SAT is almost certainly satisfiable if r 1 and almost certainly unsatisfiable if r 1. For k = 3, there has been a series of results [6, 7, 5, 13, 19, 15, 21, 11] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [15] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [21] where it was proven that a random instance of 3 SAT is almost ....

A.Z. Broder, A.M. Frieze, and E. Upfal, "On the satisfiability and maximum satisfiability of random 3-CNF formulas," Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (1993) 322--330.

....without a majority rule) and showed that for r 2:9 and r 8=3, respectively, it finds a satisfying truth assignment with positive probability. Note, that this does not imply 2:9 as a lower bound for r 3 since the algorithm succeeds only with positive probability instead of a.s. Broder et al. [3] analyzed the pure literal rule and showed that it a.s. succeeds for r 1:63, establishing r 3 1:63. Finally, Frieze and Suen [12] using some very sophisticated analysis, analyzed a generalization of uc and showed that it succeeds a.s. for r 3:003. This gives the currently best known lower ....

A.Z. Broder, and A.M. Frieze, and E. Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulae, Proceedings of the 4th ACM-SIAM Symposium on Discrete Algorithms, (1993) 322--330.

....last ten years, the satisfiability threshold conjecture has received attention in theoretical computer science, mathematics and, more recently, statistical physics. A large fraction of this attention has been devoted to the first computationally non trivial case, k = 3 and a long series of results [4, 5, 16, 1, 3, 21, 10, 22, 19, 23, 20, 11, 13] has narrowed the potential range of r 3 . Currently this is pinned between 3:42 by Kaporis, Kirousis and Lalas [21] and 4:506 by Dubois and Boufkhad [10] All upper bounds for r 3 come from probabilistic counting arguments, refining the idea of counting the expected number of satisfying truth ....

A. Z. Broder, A. M. Frieze, and E. Upfal. On the satisfiability and maximum satisfiability of random 3-CNF formulas. In Proc. 4th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 322--330, 1993.

....a threshold function c(n) such that for any # 0, as n a random formula on n variables with (c(n) #)n clauses is a.s. satisfiable, while one with (c(n) #)n clauses is a.s. unsatisfiable [Fri99] To prove an analogous result for random max k sat is much easier; this was first done by [BFU93]. Let F k (n, m) be a rnadom k sat formula on n variables with m clauses, and let f k (n, m) E(maxF k ) we may omit the subscripts k. Theorem 10. BFU93] For all k, n, c, and #, Pr( maxF k (n, cn) f k (n, cn) #) 2 exp( 2# (cn) Proof. Let X i represent the ith clause in F . ....

....(c(n) #)n clauses is a.s. unsatisfiable [Fri99] To prove an analogous result for random max k sat is much easier; this was first done by [BFU93] Let F k (n, m) be a rnadom k sat formula on n variables with m clauses, and let f k (n, m) E(maxF k ) we may omit the subscripts k. Theorem 10. [BFU93]) For all k, n, c, and #, Pr( maxF k (n, cn) f k (n, cn) #) 2 exp( 2# (cn) Proof. Let X i represent the ith clause in F . Replacing X i with an arbitrary clause cannot change maxF by more than 1. The result follows from Azuma s inequality. # Since for any fixed k and c we know that ....

Andrei Z. Broder, Alan M. Frieze, and Eli Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas, Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (Austin, TX, 1993) (New York), ACM, 1993, pp. 322-- 330. MR 94b:03023

....useful model for evaluating the effectiveness of a particular propositional proof system: strong lower bounds on proof size for random k CNF formulas attest to the fact that the proof system in question is ineffective on average. A fundamental conjecture about the random k CNF formula model, see [CS88, BFU93, CF90, CR92, FS96, KKKS98]) says that there is a constant q k , the satisfiability threshold, such that a random k CNF formula of clause density D is almost certainly satisfiable for D q k (as n gets large) and almost certainly unsatisfiable if D q k . There is considerable empirical and analytic evidence for this. ....

A. Broder, A. Frieze, and E. Upfal. On the satisfiability and maximum satisfiability of random 3-CNF formulas. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, January 1993.

....literal, irrespective of the size of the clause where it appears, counts towards the degree. Previously, Davis Putnam algorithms of increasing sophistication were rigorously analyzed towards showing that random formulas up to a certain density are a.s. as n approaches infinity) satisfiable (see [8, 9, 7, 13, 1, 5]) The best lower bound for the satisfiability threshold thus obtained is 3.26 [5] These algorithms (with the exception of the plain pure literal algorithm, which succeeds for formulas of density up to 1.63 [7, 16] take into account the size of the clauses were the literal to be set appears. ....

....certain density are a.s. as n approaches infinity) satisfiable (see [8, 9, 7, 13, 1, 5] The best lower bound for the satisfiability threshold thus obtained is 3.26 [5] These algorithms (with the exception of the plain pure literal algorithm, which succeeds for formulas of density up to 1. 63 [7, 16]) take into account the size of the clauses were the literal to be set appears. We describe here a greedy Davis Putnam algorithm that in selecting the literal to be set takes into account only how many clauses, irrespective of their size, will be satisfied. Its probabilistic analysis is based on ....

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A. Broder, A. Frieze, and E. Upfal, "On the satisfiability and maximum satisfiability of random 3-CNF formulas," Proc. 4th ACM-SIAM Symposium on Discrete Algorithms (SODA '93), pp. 322--330, 1993.

....Purdom, and Brown [16] Purdom and Brown [20] Bugrara, Pan, and Purdom [4] Franco [11] etc. study aspects of DPP for SAT using another distribution, the constant density model. A lot of the work in the constant density model involves the pure literal rule. Recently Broder, Freize, and Upfal [2] used another model to study the pure literal rule for 3 clauses. In this model there are also m k clauses, but the km literals are chosen uniformly and independently from the set of 2n available literals. A pure literal for a formula in CNF is a literal l which occurs in at least one clause, but ....

....0 1 t m 2 4N t exp Gamma m 2 2N : By the power series expansion of exp(x) this is asymptotic to exp m 2 4N exp Gamma m 2 2N ; which is asymptotic to exp Gamma 1 4 2 : Proposition 3. 8 is in striking contrast to Broder, Frieze, and Upfal s [2] results for the pure literal rule for 3 clauses. They show that in their model with asymptotically 1 probability there is no pure literal block provided m n is sufficiently small ( 1:63) Proposition 3.8 and Erdos and Renyi s [9] results on the occurrences of cycles in random graphs suggest ....

A.Z. Broder, A.M. Frieze, and E. Upfal. On the satisfiability and maximum satisfiability of random 3-CNF formulas. In Proc. 4th Annual ACMSIAM Symposium on Discrete Algorithms, pp.322-330, ACM, New York 1993.

.... Gamma ffl)n) 1; f( c ffl)n) 0. This was shown to be true for k = 2 with the constant c = 1, see [10] 17] but for k 3 was not known. For k = 3 a series of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c = 4:2: see [19] 11] 9] [8], 16] 22] 21] We now show that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. I would like to thank Svante Janson for pointing out the following subtlety to me: What I actually show is not the existence of a constant c but of a function c(n) ....

A.Z. Broder, A. M. Frieze, E. Upfal (1993) On the Satisfiability and Maximum Satisfiability of Random 3-CNF Formulas. In Proc. 4th. Ann. ACM-SIAM Symposium on Discrete Algorithms, pp322-330.

....an instance of SAT. When k = 2 a sharp threshold has been proved in [9, 15] A random instance of 2 SAT is almost surely satisfiable if r 1 and almost surely unsatisfiable if r 1. For k = 3, such a sharp threshold behaviour is widely conjectured to exist and there has been a series of papers [6, 7, 5, 12, 18, 14, 19] narrowing the area for which we do not have almost surely (un)satisfiability. The best bounds currently known come from [14] where it was proven that a random instance of 3 SAT is almost surely satisfiable if r 3:003, and [19] where it was proven that a random instance of 3 SAT is almost surely ....

A.Z. Broder, A.M. Frieze, and E. Upfal, "On the satisfiability and maximum satisfiability of random 3-CNF formulas," in Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, (1993) 322--330.

.... for carrying out the analysis of several previously studied random processes of interest, including the random loss resilient codes introduced in [9] the greedy algorithm for matchings in random graphs studied in [7] and the threshold for solving random k SAT formula using the pure literal rule [4]. In addition, generalization of these problems not previously analyzed can now be analyzed in a straightforward manner. For example, we can analyze generalizations of the processes considered in [9] As another example, we can analyze the behavior of the pure literal rule on random k SAT formula ....

....the range [ 0 ; 1) Noting that y 0 = 0 ) For a uniformly chosen k SAT formula, x) exp( Gamma (x Gamma 1) and ae(x) x k Gamma1 . By using Maple, one can verify that Condition 12 is satisfied for all y when k = 3 at the threshold value = 1:63. This result has been found previously by [4] and [11] using a different approach. The advantage of the tree analysis approach employed in this paper is that it is easily possible to analyze substantially different distributions for choosing the formula with little additional difficulty. 10 4 Greedy Matching Analysis In the paper [7] the ....

A. Broder, A. Frieze, E. Upfal, `On the Satisfiability and Maximum Satisfiability of Random 3-CNF Formulas", Proc. of the 4 th ACM-SIAM Symp. on Discrete Algorithms, 1993, pp. 322-330.

....counting arguments. The best current bound r 3 6 4:596 is due to Janson, Stamatiou and Vamvakari [15] while recently Dubois, Boufkhad and Mandler [8] announced r 3 6 4:506. Lower bounds, on the other hand, have been algorithmic in nature. The first was given by Broder, Frieze and Upfal [5] who proved that w.h.p. the pure literal heuristic succeeds on F 3 (n; rn) for r 6 1:63, but fails for r 1:7. Even before [5] Chao and Franco [6] had shown that algorithms UC and UCWM (to be defined in Section 2) find a satisfying truth assignment with positive probability for r 8=3 and r ....

....Dubois, Boufkhad and Mandler [8] announced r 3 6 4:506. Lower bounds, on the other hand, have been algorithmic in nature. The first was given by Broder, Frieze and Upfal [5] who proved that w.h.p. the pure literal heuristic succeeds on F 3 (n; rn) for r 6 1:63, but fails for r 1:7. Even before [5], Chao and Franco [6] had shown that algorithms UC and UCWM (to be defined in Section 2) find a satisfying truth assignment with positive probability for r 8=3 and r 2:9, respectively. Today, combining such results with the corollary of Theorem 1 gives lower bounds for r 3 . To improve over r ....

A. Z. Broder, A. M. Frieze, and E. Upfal. On the satisfiability and maximum satisfiability of random 3-CNF formulas. In 4th Annual ACM-SIAM Symp. on Disc. Alg. (Austin, TX, 1993), pages 322--330. ACM, New York, 1993.

....formula, if one generates less then n binary clauses the formula is almost certainly satisfiable, whereas when generating more than n clauses the formula is almost certainly unsatisfiable. For kSAT, with k 3, no such result has yet been obtained. Some preliminary results are known for 3SAT. Broder et al. 1993) show that for random 3SAT instances with a ratio of variables to clauses of less than 1.7 almost all formulas are satisfiable. Moreover, for a ratio of 5.2 or higher, it has been shown that almost all formulas are unsatisfiable (Dubois 1991; Chvatal and Szemeredi 1988) Note that these ratios are ....

Broder, A., Frieze, A., and Upfal, E. (1993). On the Satisfiability and Maximum Satisfiability of Random 3-CNF Formulas. Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, 1993, 322--330.

....useful model for evaluating the effectiveness of a particular propositional proof system: strong lower bounds on proof size for random k CNF formulas attest to the fact that the proof system in question is ineffective on average. A fundamental conjecture about the random k CNF formula model, see [CS88, BFU93, CF90, CR92, FS96, KKKS98]) says that there is a constant q k , the satisfiability threshold, such that a random k CNF formula of clause density D is almost certainly satisfiable for D q k (as n gets large) and almost certainly unsatisfiable if D q k . There is considerable empirical and analytic evidence for this. ....

A. Broder, A. Frieze, and E. Upfal. On the satisfiability and maximum satisfiability of random 3-CNF formulas. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, January 1993.

....is required and how many unsatisfiable clauses on optimal is required For such a problem, we aim to analyze behavior of the number of unsatisfiable clauses. In this paper, we consider the expected number of unsatisfiable clauses. On the expected number of satisfiable clauses, Broder et al.[1] proved the probability that the di#erence from the expected maximum number of satisfiable clause is larger than # 2m log m is less than 1 m by using Martingale technique. But they did not determine the actual expected number. We prove two lower bounds of the expected number of the unsatisfiable ....

Andrei Z. Broder, Alan M. Frieze and Eli Upfal, On the Satisfiability and Maximum Satisfiability of Random 3-CNF Formulas, in Proc. SODA'93, pp. 322-330.

....case where K = 2, Chv atal and Reed proved that the crossover point occurs when N=P = 1. In the case where K = 3, results have been much harder to obtain, but a great deal of progress has already been made. Broder et al. proved that these problems are almost certainly satisfiable when N=P 3:003 [BFU93] and Kamath et al. proved that they are almost certainly unsatisfiable when N=P 4:8 [KMPS94] Nevertheless, no one has yet been able to prove that a crossover point even exists [BFU93] 2.3 Summary This chapter has presented some previous experimental and analytical results pertaining to ....

....already been made. Broder et al. proved that these problems are almost certainly satisfiable when N=P 3:003 [BFU93] and Kamath et al. proved that they are almost certainly unsatisfiable when N=P 4:8 [KMPS94] Nevertheless, no one has yet been able to prove that a crossover point even exists [BFU93] 2.3 Summary This chapter has presented some previous experimental and analytical results pertaining to SAT. It is gratifying to observe that experimental research on SAT has influenced analytical research, and vice versa. This is specifically true for at least two important subproblems in SAT ....

A. Z. Broder, A. M. Frieze, and E. Upfal. On the satisfiability and maximum satisfiability of random 3-CNF formulas. In Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 322--330, 1993.

....be described as less hard . Note that this lower bound does not apply to algorithms that are based on cutting planes or ROBDDs [25,23] It is also known that the probabilty crossover is not the only phase transition involving random 3 SAT and that phase transitions can be solver dependent. In [7,31], the authors proved linear median running time of the pure literal algorithm at the lowdensity region (below 1.63) and showed a phase transition at 1.63 for this algorithm. In [18] the authors proved a linear median running time for another heuristic algorithm for low density instances and ....

....Figure 5 for median running time for instances of density 2, where for order above 400 the behavior is quadratic. Thus we seem to have a phase transition, corresponding to a shift from linear to quadratic behavior, between densities 1 and 2. This may coincide with the phase transition proved in [7,31] around density 1.63, as described in Section 2. At densities 4.0 and above, the median running time is exponential in the order, i.e. it behaves as 2 n , where the exponent depends on d. As with GRASP, a phase transition seems to occur between densities 3.6 and 4.0. It corresponds to the ....

A. Z. Broder, A. M. Frieze, and E. Upfal. On the satisfiability and maximum satisfiability of random 3-CNF formulas. In Proc. 4th Annual ACM-SIAM Symp. on Discrete Algorithms, pages 322--330, 1993.

....that a sequence of events En occurs almost certainly if Pr[E n ] tends to 1 as n 1. For k = 2, a sharp threshold was proved in [10, 17] a random instance of 2 SAT is almost certainly satisfiable if r 1 and almost certainly unsatisfiable if r 1. For k = 3, there has been a series of results [7, 8, 6, 14, 20, 16, 22, 12] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [16] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [22] where it was proven that a random instance of 3 SAT is almost ....

A.Z. Broder, A.M. Frieze, and E. Upfal, "On the satisfiability and maximum satisfiability of random 3-CNF formulas," Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (1993) 322--330.

....that a sequence of events En occurs almost certainly if Pr[E n ] tends to 1 as n 1. For k = 2, a sharp threshold was proved in [11, 19] a random instance of 2 SAT is almost certainly satisfiable if r 1 and almost certainly unsatisfiable if r 1. For k = 3, there has been a series of results [8, 9, 7, 15, 22, 17, 25, 13] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [17] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [25] where it was proven that a random instance of 3 SAT is almost ....

A.Z. Broder, A.M. Frieze, and E. Upfal, "On the satisfiability and maximum satisfiability of random 3-CNF formulas," Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (1993) 322--330.

.... IN 47405 USA November 9, 1999 Abstract Many of the proofs of lower bounds on the conjectured satisfiability threshold value, c , for 3SAT are based on probabilistic analyses of iterative Davis Putnam style algorithms (IDPS algorithms) or slight modifications of IDPS algorithms [4] 5] [1], 9] Let PSATA (m; n; 3) denote the probability that algorithm A finds a satisfying truth assignment for a randomly generated instance (by the fixed clause length model) where m;n are the number of clauses and variables, respectively. The probabilistic analyses proceed by finding a t 1 0 such ....

....the set of all unassigned literals. 2 Chao and Franco [4] and Chvatal and Reed [5] prove a lower bound of 1 on the conjectured value of c by probabilistically analyzing an IDPS algorithm, GUNIT, which uses as its underlying heuristic the generalized unit clause rule. Broder, Upfal, and Frieze [1] improve the lower bound to 1.63 by analyzing a slight modification of the IDPS algorithm, PURE, which uses the pure literal rule as its underlying heuristic. Broder, et al. consider the algorithm which assigns to all pure literals at each iteration. Further, once an iteration is reached in which ....

A. Z. Broder, A. M. Frieze, E. Upfal, 1993, On the Satisfiability and Maximum Satisfiability of Random 3-CNF Formulas Proceedings of the Forth ACM-SIAM Symposium on Discrete Algorithms pg. 322-330.

....3 the story is very di#erent. It is now known that a threshold for satisfiability exists in some (not completely satisfactory) sense, Friedgut [10] There has been considerable work on trying to find estimates for this threshold in the case k = 3 Chao and Franco [5, 6] Broder, Frieze and Upfal [4], Frieze and Suen [12] Achlioptas [1] Achlioptas and Sorkin [2] the last mentioned paper giving a lower bound of 3.26. Upper bounds have been pursued with the same vigour Kirousis, Kramakis, Krizanc and Stamatiou [15] Janson, Stamatiou and Vamvakari [14] Dubois, Boufkhad and Mandler [8] ....

A.Z. Broder, A.M. Frieze and E. Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas, Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, (1993) 322--330.

....the story is very different. It is now known that a threshold for satisfiability exists in some (not completely satisfactory) sense, Friedgut [10] There has been considerable work on trying to find estimates for this threshold in the case k 3 Chao and Franco [5, 6] Broder, Frieze and Upfal [4], Frieze and Suen [12] Achlioptas [1] Achlioptas and Sotkin [2] the last mentioned paper giving a lower bound of 3.26. Upper bounds have been pursued with the same vigour Kirousis, Kramakis, Krizanc and Stamatiou [15] Janson, Stamatiou and Vamvakari [14] Dubois, Boufkhad and Mandler [8] ....

A.Z. Broder, A.M. Frieze and E. Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas, Proceedings of the Fourth Annual A CM-SIAM Symposium on Discrete Algorithms, (1993) 322-330.

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A.Z. Broder, A.M. Frieze, E. Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas, Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, Austin, TX 1993.

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A.Z. Broder, A.M. Frieze, and E. Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas, Proceedings of the 4th Annual ACMSIAM Symposium on Discrete Algorithms (1993) 322-330.

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Andrei Z. Broder, Alan M. Frieze, and Eli Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas, Proc. 4th Annual Symposium on Discrete Algorithms, 1993, pp. 322--330.

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Andrei Z. Broder, Alan M. Frieze, and Eli Upfal, On the satisfiability and maximum satisfiability of random 3-CNF formulas, 4th Annual ACM-SIAM Symposium on Discrete Algorithms (Austin, TX,

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